I did not realize the body was polar or that the neck was mahogany. I Tele/strat hybrid with a poplar body. An under appreciated hardwood IMO. It's also usually the cheapest hardwood I can get at big box stores...which may contribute to my fascination. I saw something, on YT I think, about how to stain poplar to look like mahogany. This is not that video...but it's a similar idea.
I'll see if I can find the video. Meanwhile since you live in the US you can probably get some cheap pieces of poplar at HD or Lowes to practice matching...if that's the route you ultimately go. This post is strictly in the FWIW category. I don't think I'd be inclined to do anything other than clear-coat or paint that side for fear of disturbing what you've already painted.